Answer
Divergent
Work Step by Step
By Ratio Test, we have
$\lim_{n\to\infty}\biggl| \frac{(n+1)5^{2n+2}}{10^{n+2}}\frac{10^{n+1}}{n5^{2n}}\biggr| =\lim_{n\to\infty}\biggl|\frac{25}{10}\frac{n+1}{n}\biggr| = \frac{25}{10}$
Since the limit $ > 1$, the series is divergent by Ratio Test